# how to solve this system of equations with a parameter

I am a selflearner and really new to mathematica software and I am struggling to solve the following system of equations...:

$$(x-d)(y-d)+3(x+1)(1-y)=0$$

$$(x-d)^2+3(x+1)^2=(y-d)^2+3(1-y)^2$$

$$(z-x)(y-d)+(w-x\sqrt3-\sqrt3)(-y\sqrt3+\sqrt3)=0$$

$$(y-d)\frac{(1+x)\sqrt3+w}{2}+\sqrt3(y-1)\frac{x+z-2d}{2}=0$$

$$\vert{x}\vert,\vert{y}\vert, \vert{z}\vert, \vert{d}\vert<1,0

The $$x, y,z, w$$ are the unknown quantities and $$d$$ a parameter (all real). Probably there is only one solution iff $$d$$ is restricted to a smaller interval than it is defined...

Every time I write the code in the notebook I get the message "WolframAlpha cannot understand what you want to do". Could someone help me please?

What I write in the notebook is:

Reduce[{(x-d)(y-d)+3(x+1)(1-y)==0, (x-d)^2+3(x+1)^2==(y-d)^2+3(1-y)^2, (z-x)(y-d)+(w-xsqrt3-sqrt3)(-ysqrt3+sqrt3)==0, (y-d)(1/2)((x+1)sqrt3+w)+(y-1)sqrt3(1/2)(x+z-2d)==0, -1

• Can you include the code that you tried? Commented Sep 6, 2019 at 19:59
• @MelaGo, I edited my question as you consulted me and sorry for the late response but I had to do some things in real life here ): Commented Sep 7, 2019 at 11:03

Clear["Global*"]

eqns = And @@ {(x - d) (y - d) + 3 (x + 1) (1 - y) == 0,
(x - d)^2 + 3 (x + 1)^2 == (y - d)^2 + 3 (1 - y)^2,
(z - x) (y - d) + (w - x Sqrt[3] - Sqrt[3]) (-y Sqrt[3] + Sqrt[3]) == 0,
(y - d) ((1 + x) Sqrt[3] + w)/2 + Sqrt[3] (y - 1) (x + z - 2 d)/2 == 0,
-1 < x < 1, -1 < y < 1, -1 < z < 1,
0 < w < Sqrt[3], -1 < d < 1};


The equations as you expressed them have no solution

Solve[eqns, {x, y, z, w}, Reals]

{}


Relaxing the constraint on z, w, and d to merely that they are real

solR = Solve[eqns[[1;;6]], {x, y, z, w}, Reals];


There is a single solution

Length@solR

(* 1 *)


The variables are each expressed as a ConditionalExpression with a common condition

cond = solR[[1, All, -1, -1]] // Union

(* {-2 + Sqrt[3] < d < 2 - Sqrt[3]} *)


which is approximately

cond[[1]] // N

(* -0.267949 < d < 0.267949 *)


Using this constraint

sol = solR[[1]] // FullSimplify[#, cond] &

(* {x -> 1/2 (-3 + Sqrt[3] - (1 + Sqrt[3]) d),
y -> 1/2 (3 - d - Sqrt[3] (1 + d)), z -> 1/2 (3 + Sqrt[3] (-1 + d) + 5 d),
w -> 1/2 (-3 + Sqrt[3] + (3 + Sqrt[3]) d)} *)


Verifying,

eqns[[1 ;; 6]] /. sol // Simplify[#, cond] &

(* True *)


The intervals for {x, y, z, w} are

({x, y, z, w} /. sol // Simplify) /.
d -> Interval[{-2 + Sqrt[3], 2 - Sqrt[3]}] // Simplify

(* {Interval[{-1, -2 + Sqrt[3]}], Interval[{2 - Sqrt[3], 1}],
Interval[{-2 + Sqrt[3], 5 - 2 Sqrt[3]}], Interval[{-3 + Sqrt[3], 0}]} *)


or approximately,

% // N // Chop

(* {Interval[{-1., -0.267949}], Interval[{0.267949, 1.}],
Interval[{-0.267949, 1.5359}], Interval[{-1.26795, 0}]} *)
`
• Thanks a lot for your answer, I did a mistake in one of the equations I wrote in my questions but I understood the way you solved it so I will do it myself now, +25 Commented Sep 7, 2019 at 17:12